Constrained particle on a group: from propagators to correlators

Abstract

We develop a particle-on-a-group formulation of super-JT gravity aimed at computing supersymmetric correlators. We show that the (super)JT gravity can be described by a particle moving on the isometry group satisfying constraints from boundary conditions of (super)JT gravity. In this language the $\mathcal N=2$ and $\mathcal N=4$ theories are described by constrained particles on $SU(1,1|1)$ and $PSU(1,1|2)$. Solving the constraints gives the super-Schwarzian actions. We also quantize the reduced superparticle, with careful treatment of the fermionic constraints. We then derive the physical worldline supercharges from the requirement that the transformations preserve the constraints. These charges allow us to construct supersymmetric interval propagators in invariant variables and to formulate boundary-anchored Wilson-line operators for both superconformal primary and descendant insertions. Finally, we use these ingredients to build an algorithm for correlators. We obtain the $\mathcal N=2$ and $\mathcal N=4$ three-point composition kernels and zero-energy scalar three-point functions. For four-point functions, the same method reproduces the standard bosonic JT OTOC, gives an explicit zero-energy OTOC in $\mathcal N=2$ and $\mathcal N=4$ SJT, which is potentially useful for studying Berry curvature and BPS chaos.